Past Events
We will derive the transport equation as the second term in our approach to solving hyperbolic PDEs, describe the meaning of this equation from our symplectic perspective, and, if time permits, outline a solution strategy.
Steady Kahler-Ricci solitons are eternal solutions of the Kahler-Ricci flow. I will present new examples of such solitons with strictly positive sectional curvature that live on C^n and provide an answer to an open question of H.-D. Cao in complex dimension n>2. This is joint work with Pak-…
I will talk about a new algebra of operations on polynomials which has the property
$T_iT_j=T_jT_{i+1}$ for $i>j$ and a family of polynomials dual to them called forest polynomials. This family of operations plays the exact role for quasisymmetric polynomials and forest polynomials as…
We survey some extensions of the classical notions of Du Bois and rational singularities, known as the k-Du Bois and k-rational singularities. By now, these notions are well-understood for local complete intersections (lci). We explain the difficulties beyond the lci case, and propose new…
We will give an accessible introduction to the Hasse-Weil conjecture for curves and related problems in the Langlands program. We will discuss some recent work with G. Boxer, F. Calegari and T. Gee in which we prove the modularity of a positive proportion of genus two curves. We will explain…
We count squarefree numbers in short intervals [X, X+H] for H > X^{1/5 - \delta}, where \delta > 0 is some absolute constant. This improves on the exponent 1/5 shown by Filaseta and Trifonov in 1992.
In improving bounds on the number of integers in a short interval divisible by…
Abstract: Inspired by a recent breakthrough work of Gorodetsky, Matomaki, Radziwill and Rodgers on variance of squarefree numbers in short intervals, a similar study for variance of squarefull numbers in short intervals was carried out. In this talk, I will highlight some of the journeys in this…
This talk discusses the unstructured sparse recovery problems of a general form. The task is to recover the spike locations and weights of an unknown sparse signal from a collection of its unstructured observations. Examples include rational approximation, spectral function estimation, Fourier…
Given n points and a smooth Jordan curve in the complex plane, what is the minimum degree of a non-constant polynomial which maps all of the points to the curve? It is easy to bound the degree above by n-1, while if the points are collinear and the curve is an ellipse, then the degree is…
Abstract: In this work we prove global well-posedness for the massive Maxwell-Dirac equation in the Lorentz gauge in $\mathbb{R}^{1+3}$, for small and localized initial data, as well as modified scattering for the solutions. In doing so, we heuristically exploit the close connection…